Dow Jones Industrial Average is:
“a stock index that tracks 30 of the largest U.S. companies. Created in 1896, it is one of the oldest stock indexes, and its performance is widely considered to be a useful indicator of the health of the entire U.S. stock market”.
(source of definition: https://www.fool.com/investing/stock-market/indexes/dow-jones/;
source of data: https://www.wsj.com/market-data/quotes/index/DJIA/historical-prices )
FTSE China A50 is:
“an index for 50 stocks of companies with the highest market capitalisation listed on the Shanghai and Shenzhen stock exchanges”.
(source of definition: https://www.avatrade.com/trading-info/financial-instruments-index/indices/china-a50;
source of data: https://www.wsj.com/market-data/quotes/index/XX/XIN9/historical-prices )
FTSE 100 is:
“an index composed of the 100 largest (by market capitalisation ) companies listed on the London Stock Exchange (LSE)”.
(source of definition: https://www.ii.co.uk/knowledge-centre/quick-guides/before-you-start/what-is-the-ftse-100;
source of data: https://www.wsj.com/market-data/quotes/index/UK/UKX/historical-prices)
NASDAQ is:
“The first electronic stock market listing over 5000 companies. The Nasdaq stock market comprises two separate markets, namely the Nasdaq National Market, which trades large, active securities and the Nasdaq Smallcap Market that trades emerging growth companies”.
(source of definition: https://www.nasdaq.com/glossary/n/nasdaq-stock-market;
source of data: https://www.wsj.com/market-data/quotes/index/NASDAQ/historical-prices)
NIKKEI 225 is:
“the most recognized Japanese stock market index. It comprises Japan’s top 225 companies that are listed on the Tokyo Stock Exchange. The Nikkei Index is considered an important measure of the Japanese stock market and the performance of the Japanese economy.”
(source of definition: https://corporatefinanceinstitute.com/resources/knowledge/trading-investing/nikkei-index/;
source of data: https://www.wsj.com/market-data/quotes/index/JP/NIK/historical-prices)
S&P 500 is:
“a market-capitalization-weighted index of 500 leading publicly traded companies in the U.S. It is not an exact list of the top 500 U.S. companies by market cap because there are other criteria that the index includes.”
(source of definition: https://www.investopedia.com/terms/s/sp500.asp;
source of data: https://www.wsj.com/market-data/quotes/index/SPX/historical-prices)
EURO STOCK 50 :
“represents the performance of the 50 largest companies among the 20 supersectors in terms of free-float market cap in Eurozone countries. The index has a fixed number of components and is part of the STOXX blue-chip index family. The index captures about 60% of the free-float market cap of the EURO STOXX Total Market Index (TMI)”
(source of definition: https://www.stoxx.com/document/Bookmarks/CurrentFactsheets/SX5GT.pdf;
source of data: https://www.wsj.com/market-data/quotes/index/XX/SX5E/historical-pricess)
VIX is:
“based on the prices of options on the S&P 500 Index and is calculated by aggregating weighted prices of the index’s call and put options over a wide range of strike prices.”
(source of definition: https://corporatefinanceinstitute.com/resources/knowledge/trading-investing/vix-volatility-index/;
source of data: https://www.wsj.com/market-data/quotes/index/VIX/historical-prices)
This plot below presents all indexes:
This interactive plot below presents value of EURO STOCK 50:
A time series is an ordered set of measurements taken at regular intervals, an ideal example of which is the stock exchange indexes.
Time series plot for EURO STOCK 50 is as follows:
Time series wad broke down into: seasonal component, trend, and residuals.
We see some seasonal character of our data.
Augmented Dickey-Fuller Test for stationarity confirms that variable EURO STOCK 50 is non-stationary:
##
## Augmented Dickey-Fuller Test
##
## data: tsSX5Euro
## Dickey-Fuller = -2.3361, Lag order = 3, p-value = 0.4438
## alternative hypothesis: stationary
Then it was done time series evaluation using autocovariance (acf function) and partial autocovariance (pacf function). Autocovariance presents the correlation of the time series with itself shifted by a certain time interval. In turn, partial autovariance is the size of the correlation between the time series and its shift (Lander, 2018, s.404).
This plot shows result of autocovariance:
This plot shows result of partial autocovariance:
The charts confirm the non-stationary nature of the trend. Therefore, a differentiation has to be performed. The number of differentiations was determined using the ndiffs function and amounted to 0.
The arima function showed that the optimal model for the discussed time model will be ARMA (1, 0, 0). The ACF and PACF for the ideal model show the white noise pattern:
This is the result of building ARIMA model(1,0,0):
## Series: tsSX5Euro
## ARIMA(1,0,0) with drift
##
## Coefficients:
## ar1 intercept drift
## 0.8939 3550.3984 4.0049
## s.e. 0.0798 127.7783 6.1955
##
## sigma^2 = 3640: log likelihood = -159.26
## AIC=326.53 AICc=328.19 BIC=332
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 5.772513 57.12563 50.13811 0.1384377 1.367824 0.2815538 0.0824524
This is prediction based on the ARIMA model forecasting for 24 months with the standard error:
Prediction was also performed with Naive Forecasting Method. This is a result:
##
## Forecast method: Naive method
##
## Model Information:
## Call: naive(y = tsSX5Euro)
##
## Residual sd: 58.9613
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 4.155 58.96128 51.23857 0.1057854 1.395605 0.2877335 0.05458368
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jul 2022 3554.8 3479.238 3630.362 3439.238 3670.362
## Aug 2022 3554.8 3447.939 3661.661 3391.371 3718.229
## Sep 2022 3554.8 3423.923 3685.677 3354.641 3754.959
## Oct 2022 3554.8 3403.676 3705.924 3323.676 3785.924
## Nov 2022 3554.8 3385.838 3723.762 3296.396 3813.204
## Dec 2022 3554.8 3369.712 3739.888 3271.732 3837.868
## Jan 2023 3554.8 3354.882 3754.718 3249.052 3860.548
## Feb 2023 3554.8 3341.079 3768.521 3227.941 3881.659
## Mar 2023 3554.8 3328.114 3781.486 3208.114 3901.486
## Apr 2023 3554.8 3315.852 3793.748 3189.361 3920.239
This is a plot:
Prediction was also performed with Holt’s Trend Method:
##
## Forecast method: Holt's method
##
## Model Information:
## Holt's method
##
## Call:
## holt(y = tsSX5Euro, h = 24)
##
## Smoothing parameters:
## alpha = 0.9999
## beta = 1e-04
##
## Initial states:
## l = 3543.8215
## b = 3.0165
##
## sigma: 65.9243
##
## AIC AICc BIC
## 346.2808 348.8895 353.1173
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -2.646898 61.20921 53.00171 -0.08589335 1.451215 0.2976345
## ACF1
## Training set 0.06696209
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jul 2022 3557.818 3473.333 3642.304 3428.609 3687.028
## Aug 2022 3560.827 3441.347 3680.308 3378.098 3743.557
## Sep 2022 3563.836 3417.498 3710.174 3340.032 3787.641
## Oct 2022 3566.845 3397.861 3735.829 3308.407 3825.283
## Nov 2022 3569.854 3380.916 3758.792 3280.898 3858.809
## Dec 2022 3572.863 3365.882 3779.843 3256.313 3889.412
## Jan 2023 3575.872 3352.296 3799.447 3233.943 3917.801
## Feb 2023 3578.880 3339.857 3817.904 3213.325 3944.436
## Mar 2023 3581.889 3328.354 3835.424 3194.141 3969.638
## Apr 2023 3584.898 3317.636 3852.161 3176.155 3993.641
## May 2023 3587.907 3307.586 3868.228 3159.193 4016.621
## Jun 2023 3590.916 3298.116 3883.716 3143.116 4038.715
## Jul 2023 3593.925 3289.153 3898.696 3127.817 4060.032
## Aug 2023 3596.934 3280.642 3913.225 3113.207 4080.660
## Sep 2023 3599.942 3272.533 3927.352 3099.213 4100.672
## Oct 2023 3602.951 3264.788 3941.115 3085.775 4120.128
## Nov 2023 3605.960 3257.372 3954.548 3072.840 4139.080
## Dec 2023 3608.969 3250.257 3967.681 3060.366 4157.572
## Jan 2024 3611.978 3243.418 3980.538 3048.314 4175.642
## Feb 2024 3614.987 3236.833 3993.140 3036.651 4193.323
## Mar 2024 3617.996 3230.484 4005.507 3025.348 4210.643
## Apr 2024 3621.004 3224.354 4017.655 3014.380 4227.629
## May 2024 3624.013 3218.428 4029.598 3003.725 4244.302
## Jun 2024 3627.022 3212.693 4041.351 2993.361 4260.683
The predictions made by all methods were compared. MAPE and MAE turned out to be the smallest in the ARIMA model.
Multiple regression model for EURO STOXX 50 Index is as follows:
##
## Call:
## lm(formula = SX5Euro_Close ~ DJ_Close + FTSE_Close + FTSE100_Close +
## NASDAQ_Close + NIKKEI_Close + SP500_Close + VIX_Close, data = all_indexes)
##
## Residuals:
## Min 1Q Median 3Q Max
## -235.225 -55.620 4.422 61.322 232.643
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.076e+02 1.124e+02 -5.407 9.72e-08 ***
## DJ_Close 1.607e-01 1.069e-02 15.027 < 2e-16 ***
## FTSE_Close -2.881e-02 5.533e-03 -5.207 2.76e-07 ***
## FTSE100_Close 4.110e-01 2.613e-02 15.732 < 2e-16 ***
## NASDAQ_Close 2.787e-01 1.773e-02 15.717 < 2e-16 ***
## NIKKEI_Close -3.225e-02 4.737e-03 -6.808 2.70e-11 ***
## SP500_Close -1.493e+00 1.118e-01 -13.351 < 2e-16 ***
## VIX_Close 2.803e+00 7.515e-01 3.730 0.000213 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 85.2 on 527 degrees of freedom
## (71 observations deleted due to missingness)
## Multiple R-squared: 0.9652, Adjusted R-squared: 0.9647
## F-statistic: 2086 on 7 and 527 DF, p-value: < 2.2e-16
The model explains 96% of the variability of EURO STOXX 50 Index, and all the variables used in the model are significant.
## -----------------------------------------------
## Test Statistic pvalue
## -----------------------------------------------
## Shapiro-Wilk 0.9938 0.0275
## Kolmogorov-Smirnov 0.0534 0.0945
## Cramer-von Mises 43.6626 0.0000
## Anderson-Darling 1.0561 0.0089
## -----------------------------------------------
The residuals in model meet the assumptions of the normal distribution, and the result of the K-S test also it confirms (p > 0,05 that is, there is no reason to reject the null hypothesis and the distribution is normal).
The Quantile-Quantile plot also confirms the normal distribution of EURO STOXX 50 Index - the observations are almost perfectly positioned on the straight line (except for the rest).
The Cook’s plot above allows you to explore outliers.
Multiple regression models were also compared. The following regression models were built:
They were visualized using the multiplot function.
## Analysis of Variance Table
##
## Model 1: SX5Euro_Close ~ DJ_Close + FTSE_Close + FTSE100_Close + NASDAQ_Close +
## NIKKEI_Close + SP500_Close + VIX_Close
## Model 2: SX5Euro_Close ~ DJ_Close + FTSE100_Close + NASDAQ_Close + SP500_Close +
## VIX_Close
## Model 3: SX5Euro_Close ~ VIX_Close + NASDAQ_Close + SP500_Close
## Model 4: SX5Euro_Close ~ VIX_Close + FTSE100_Close
## Model 5: SX5Euro_Close ~ NIKKEI_Close + FTSE_Close
## Model 6: SX5Euro_Close ~ DJ_Close
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 527 3825475
## 2 529 4645803 -2 -820328 56.505 < 2.2e-16 ***
## 3 531 7358661 -2 -2712859 186.863 < 2.2e-16 ***
## 4 532 14208455 -1 -6849794 943.632 < 2.2e-16 ***
## 5 532 16145983 0 -1937528
## 6 533 6057648 -1 10088335
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The smallest RSS is observed in the case of model 1. This is also confirmed by the results of AIC ( Akaike Information Criterion) and BIC ( Bayesian Information Criterion).
Due to the obsolescence of using ANOVA to test regression models (Lander, 2018, p. 337), a cross-validation with generalized linear models was also performed.
## Error Adjusted Error model_name
## 1 7342.845 7321.294 modelG1
## 2 9048.385 9007.159 modelG2
## 3 14073.156 14036.490 modelG3
## 4 27326.747 27240.330 modelG4
## 5 30497.247 30462.139 modelG5
## 6 11430.562 11418.570 modelG6
It has been shown once again that the first model is characterized by the lowest error value.
The correlation between the index EURO STOCK 50 and GDP in Europe was also checked. For this purpose, the data was converted into quarterly data. Data on quarterly GDP was downloaded from the website: https://ec.europa.eu/eurostat/databrowser/view/NAIDQ_10_GDP/default/table?lang=en
Correlation between EURO STOCK 50 and GDP is 0.8323101 - it is a high correlation.
In summary, we see fertility rate and birth rate falling over the years, while death rate and life expectancy are increasing. It is very important to follow demographic changes in countries and individual parts of the world so that governments can react early and shape their policies on public health, pro-family policies, etc.It is impossible to track demographic changes in isolation from the data on armed conflicts and the related migration of people, climate change, and the pandemic situation.
References: